Welcome back to Linear Circuits, where,
today, we're doing
Part 2 of our Power Factors and Power
Triangles lecture.
In the previous lesson, we talked about
calculating Complex Power
and identifying what the various aspects
of that complex power represented.
Today, we're going to continue on talking
about the
power factors and power triangles, most,
particularly, about power triangles.
And as we understand those things, we'll
then be able to talk more about maximum
power transfer in AC systems.
The objectives of this lesson are to have
you use power triangles and to calculate
power angle and power factor, real and
Reactive Power, and Apparent Power for a
system.
When calculating the Complex Power we
plotted it on the complex plane.
Where we had the real part here, and the
imaginary part here.
And we label all of the various values.
What we're going to do is we're going
to take the triangle out of this complex
plane,
and draw it like this.
We see that we have all the same basic
measurements.
We have the P, the Q, and the absolute
value of S, corresponding to
Real Power, Reactive Power, and the Parent
Power, and our angle theta, our power
angle.
So to remind us of the equation we use for
finding the
Complex Power, I've listed it here and
I've put the power triangle.
And, what we're going to do is find out
how to calculate all these things from the
equations.
So, here we have the voltage equation, and
the current equation would be m cosine
of omega t, plus theta v, and Im cosine of
omega t plus theta i.
Now we'll go through, one by one, all of
the different
important things that we're going to be
defining for these power triangles.
First of all, the power angle.
Power angle theta, is the single right
here.
And
that corresponds to theta v minus theta i,
which we
can pull immediately from our equations
for voltage and current.
We're going to call the cosine of theta
the power factor.
And the reason we'll say that is that we
can calculate the length of P, given we
know, theta, the Apparent Power, absolute
value of S,
as P is equal to modulo S, times cosine
of theta.
And so that's why we called it the power
factor because it's how much
we're multiplying by our Apparent Power to
find the average power that's being
consumed.
So that means that we can calculate
Average Power by taking the RMS voltage.
Multiplying it by the RMS current and
multiplying that by cosine theta.
And this just follows immediately.
It consists of our calculation of how we
found the Apparent Power.
Then for Reactive Power, Q, we want to
find the length of this leg.
And using, again, basic trigonometry, Q is
going to be
equal to the Apparent Power times the sine
of theta.
Now,
here's where it gets a little bit hairy as
far as units are concerned.
Average power is going to be measured in
watts, because we're doing work.
And watts is the rate at which energy is
being used.
But, when we're talking about Reactive
Power, it corresponds to power
that's temporarily being stored up to be
used at a later time.
And it's not actually doing any real work.
So to help distinguish it, we're going to
be using a different unit.
The AC units are Volt-Amperes Reactive.
Sometimes distinguished as V-A-R-S, or
VARS.
It's basically measuring the same thing
because we're taking a voltage.
Multiplying it by a current and
then multiplying it by something that's
unitless.
which is basically the same thing we have
for power.
And so we're just going to call
it something different to help distinguish
that
it's not really being used, it's just a
temporary storage type of a thing.
And Apparent Power we're again going to
use a different unit.
We're going to use Volt-Amperes or VA, and
we've
already discussed how to calculate that
Apparent Power.
So now we see that we can calculate all
these different values, from the,
the voltage and the current equations.
And that they have a relationship that
corresponds to
the trigonometric relationship that we see
in a triangle.
Now we can look at the relationships
between the Apparent Power, the Real
Power, and the Complex Power, or
the, Reactive Power, rather for different
devices.
So, first of all, the resistive case,
we've already looked at.
The voltage and the currents multiply
together
to get something that is non-negative, for
power.
Here in the green.
And I'm still using the same v is
red, i is blue and green is P.
Again, all different units here.
I, here, illustrate the Impedance of this
device because it's all real.
It's just a real line out here.
Now, this is a triangle, in, the
generative sense, where this phase angle,
this angle here in this corner is 0.
And we all
see that the, the Impedance triangles are
similar,
in the mathematical sense of similar, to
the Complex Power triangles.
All of my power, all of my Parent Power is
Real Power and there is no Reactive Power
for a resistor.
Now, as I move this around.
Get more of a proper triangle.
Now I have a slightly inductive load,
because I have a resistor, I have an
inductor.
Now, you'll notice that, on these plots,
my voltages stay the same.
I'm moving currents with respect to the
voltages
and then, obviously, the powers are
changing as well.
No longer are the voltage and the current
in phase.
Now, the voltage, leads the current, a
little
bit, or the current lags behind the
voltage.
This is my Impedance triangle
for this system, where the real part comes
from the resistor.
The imaginary part comes from the
inductor.
So the real part and the imaginary part
are giving us this.
And this theta, corresponds to this data
because these are two similar triangles.
For systems that have, a, an inductive
load, my phase
angle is going to be between 0 and pi
halves for
my power angle.
And my power is going to be somewhere
between the
extremes of 0 and the value of the
Apparent Power.
But so is my react, my Reactive Power,
somewhere between 0 and the the Apparent
Power.
We're going to say that the system is
lagging, because it's inductive.
And we say that it lags, or it's lagging
because the current
lags behind the voltage, the voltage hits
its maximum point before the current.
We'll go to the extreme case where it is a
pure inductor.
Now my triangles are again degenerative
because
my angle, my power angles are pi halves.
There's no Real Power, real average power
being consumed.
All of the Apparent Power goes to Reactive
Power.
I can do the same thing with capacitive
loads.
Like this.
Now, we see that the current leads the
voltage, so it's going to be leading.
This is my Impedance triangle, which is
similar to my power triangle.
My theta, my power angle, is between minus
pi halves and 0.
My power is between 0 and the Apparent
Power, and
my Reactive Power is now going to be
negative because it's
down here in this quadrant.
And it's going to be between the minus
value of the Apparent Power and the 0.
We can see all of the relationships here.
And finally, looking at the purely
capacitive case,
all of the power is Reactive Power because
the end of the capacitor stores up the
energy temporarily to be used at a later
time.
What are the implications of this?
Well, first of all, only Real Power is
being transformed as
heat or light or moving a motor or doing
real work.
And, typically that's what we're most
interested in.
We don't really care about how much power
is being temporarily stored up in
the system, we want to know how much is
being used to do real work.
But that doesn't mean we can neglect the
Reactive Power.
Reactive power causes increased currents.
And so that means that the more Reactive
Power
we have in our system, the more currents
need
to be used in the lines connecting to your
device.
And so you
[INAUDIBLE],
have more loss in the transmission lines,
leading to that device.
So if you're a power company, you'll want
to give some heat to that Reactive Power.
Now, private customers, for their private
homes and residences, typically,
only are charged for the Real Power that
they consume.
The power company doesn't really care too
much about how much Reactive
Power there is because a resi, residential
home doesn't have a lot.
But if you're a big
industrial system, with big motors,
there's a lot
of inductive loads in those types of
systems.
And so consequently industrial consumers,
are often charged at a lower rate
for the Reactive Power because the power
companies needs to have more equipment.
They have more losses in their
transmission
lines and they need to design their
systems with extra transformers and extra
devices,
to be able to handle those increased
loads.
So,
[INAUDIBLE]
because of this, big industrial users
might want
to give some consideration to their
systems and
change their systems a little bit to
reduce
the amount of Reactive Power that they
have.
To reduce the costs that they pay to the
power company.
In summary, we've defined power angle and
power
factory, factor, Real and Reactive Power,
and Apparent Power.
And showed how they all correlate in a
triangle behavior,
using power triangles, we can see
the relationship between all those
different values.
In the next lesson, we'll see how we can
use reactive elements like capacitors
and inductors to control the amount of
Reactive Power that we have in the system.
And we can use, or we can analyze maximum
power transfer for AC systems, and look at
how that is different from the maximum
power
transfer that was calculated in DC
systems, until then.